Practical Filtering Methods with Model Errors

The projects in this proposal are part of the PI's long-term career goal to deliver a class of practically scalable data assimilation (or filtering) schemes with solid theoretical foundations for state estimation of geophysical fluid dynamics. This proposal is an outgrowth of the PI's recent successful effort in designing accurate, reduced filtering methods with cheap stochastic models as alternatives to expensive models. Four projects are proposed: 1. Design computationally faster stochastic filters to assimilate atmospheric infrared sounder (AIRS) in the presence of multiple cloud types in the tropics. 2. Develop stable linear autoregressive (AR) filters for nonlinear, weakly chaotic dynamical systems. This project involves designing a novel parameterization scheme for AR models that avoids utilizing a long time series as in classical regression strategy, yet respects the sufficient conditions for optimal AR filtering, established in the PI's recent work. 3. Study the role of higher order terms of the singular perturbation expansion when a reduced model from classical averaging theory is used in filtering multi-scale interaction between modes of turbulent signals with moderate separation of scales. This study involves formal asymptotic expansion and rigorous error estimation. The PI will show that the higher order terms are important to avoid covariance underestimation in the presence of model errors. 4. Develop a fast filtering framework to assimilate multi-scale dynamical systems with 'superparameterization', a fast numerical scheme to resolve interaction of cloud-scale dynamics and large-scale tropical convecting atmosphere. The new algorithm will include an online small-scale estimation scheme that imposes statistical consistency between the large and small-scale variables.

Fundamental issues in real-time weather prediction are model errors. This problem is attributed to incomplete understanding of the physics and our lack of computational resources to resolve physical processes in various time and length scales. Modern operational weather models poorly reproduce the tropical observational records even after resolving 10 billion variables. This long-standing issue prevents the global weather model forecasting skill to improve from weekly to monthly, as reported in a recent article in the World Meteorological Organization bulletin. The results from this proposal will transform the future design of computational methods for various prediction related problems in the presence of model errors, in particular numerical weather prediction. This proposal supports an interdisciplinary research training environment for a graduate student, involving mathematical analysis, statistical modeling, and scientific computing. The PI, who is jointly appointed as a faculty in the mathematics and meteorology departments at PSU, will develop an interdisciplinary graduate course with emphasis on PDE and waves for atmospheric and ocean modeling.